Smooth Shape-Aware Functions with Controlled Extrema

Shape deformation: Recent works emphasize the importance of bounded control,
but simply adding constant bounds to shape-aware smoothness energies of
increasing order encourages more and more oscillation. Our framework
efficiently optimizes such high-order energies as ∫M
‖∇ 4f ‖ while ensuring against
spurious local extrema.

abstract

Functions that optimize Laplacian-based energies have become popular in
geometry processing, e.g. for shape deformation, smoothing, multiscale kernel
construction and interpolation. Minimizers of Dirichlet energies, or solutions
of Laplace equations, are harmonic functions that enjoy the maximum principle,
ensuring no spurious local extrema in the interior of the solved domain occur.
However, these functions are only C0 at the constrained points, which often
causes smoothness problems. For this reason, many applications optimize
higher-order Laplacian energies such as biharmonic or triharmonic. Their
minimizers exhibit increasing orders of continuity but also increasing
oscillation, immediately releasing the maximum principle. In this work, we
identify characteristic artifacts caused by spurious local extrema, and provide
a framework for minimizing quadratic energies on manifolds while constraining
the solution to obey the maximum principle in the solved region. Our framework
allows the user to specify locations and values of desired local maxima and
minima, while preventing any other local extrema. We demonstrate our method on
the smoothness energies corresponding to popular polyharmonic functions and
show its usefulness for fast handle-based shape deformation, controllable color
diffusion, and topologically-constrained data smoothing.